The Kirchhoff theory of thin plates and the Kirchhoff-Love theory of thin shells are characterized by energy functionals which depend on curvature, consequently they contain second-order derivatives of displacement. The resulting Euler-Lagrange—or equilibrium—equations in turn take the form of fourth-order partial differential equations. It is well-known from approximation theory that in this context, the convergence of finite-element solutions requires so-called C1 interpolation. More precisely, in order to ensure that the bending energy is finite, the test functions have to be H2, or square-integrable functions whose first- and second-order derivatives are themselves square-integrable. Unfortunately, for general unstructured meshes it is not possible to ensure C1 continuity in the conventional sense of strict slope continuity across finite elements when the elements are endowed with purely local polynomial shape functions and the nodal degrees of freedom consist of displacements and slopes only. Inclusion of higher-order derivatives among the nodal variables leads to well-known difficulties, e.g., the inability to account for stress and strain discontinuities in shells whose properties vary discontinuously across element boundaries, and, owing to the high order of the polynomial interpolation required, the presence of spurious oscillations in the solution.
The difficulties inherent in C1 interpolation have motivated a number of alternative approaches, all of which endeavor to “bear” the C1 continuity requirement. Examples are: quasi-conforming elements obtained by relaxing the strict Kirchhoff constraint; the use of Reissner-Mindlin theories for thick plates and shells (which requires conventional C0 interpolation only); reduced-integration penalty methods; mixed formulations; degenerate solid elements; and many others known from the literature. C0 elements often exhibit poor performance in the thin-shell limit—especially in the presence of severe element distortion. Such distortion may be due to a variety of pathologies such as shear and membrane locking. The proliferation of approaches and the rapid growth of the specialized literature attest to the inherent, perhaps insurmountable, difficulties in vanquishing the C1 continuity requirement.